Problem 35 - Calculus III

Question

Consider the curve described by the vector-valued function \(\mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k}\). [T] Use technology to sketch the curve.

Step-by-Step Solution

Answer

Input the function in a 3D graphing tool to plot and observe the curve.

Step 1: Understanding the Vector-Valued Function

The given vector-valued function is \( \mathbf{r}(t)=\left(50 e^{-t} \cos t\right) \mathbf{i}+\left(50 e^{-t} \sin t\right) \mathbf{j}+\left(5-5 e^{-t}\right) \mathbf{k} \). This function describes a three-dimensional curve, where \(x(t) = 50 e^{-t} \cos t\), \(y(t) = 50 e^{-t} \sin t\), and \(z(t) = 5 - 5 e^{-t}\). We need to sketch this curve using technology such as a graphing software or online plotting tool.

Step 2: Setting Up the Graphing Tool

Choose a graphing software such as Desmos, GeoGebra, or an online 3D graphing tool. Input the parametric equations derived from \( \mathbf{r}(t) \): \(x = 50 e^{-t} \cos t\), \(y = 50 e^{-t} \sin t\), and \(z = 5 - 5 e^{-t}\). Set the parameter \(t\) to an appropriate range, such as \(t \in [0,10]\), to clearly observe the behavior of the curve.

Step 3: Plotting the Curve

Execute the plotting function in the graphing tool. Use the equations from Step 2 to generate the three-dimensional curve. Adjust the viewing angle or zoom level if necessary to get a clear view of how the curve behaves in three-dimensional space.

Step 4: Analyzing the Produced Sketch

Once the curve is plotted, observe how the components \(x(t)\), \(y(t)\), and \(z(t)\) contribute to its shape. The terms \(e^{-t}\) cause the spiral effect, making the curve approach a point as \(t\) increases, due to the exponential decay. The \(z\)-component \(5 - 5e^{-t}\) indicates the curve starts near \(z=0\) and asymptotically approaches \(z=5\).

Key Concepts

Parametric Equations

Parametric equations offer a way to represent curves and surfaces in a mathematical space by expressing the coordinates of points as functions of a variable, typically denoted as \( t \). In the given vector-valued function, we see a set of parametric equations:

• \( x(t) = 50e^{-t} \cos t \)
• \( y(t) = 50e^{-t} \sin t \)
• \( z(t) = 5 - 5e^{-t} \)

Each equation accounts for one of the spatial dimensions \( x, y, \) and \( z \). By varying the parameter \( t \), different points on the curve are generated.
Parametric equations help us understand complex curves and motions. Different parametric forms can articulate distinct paths, not only in two dimensions but also in three-dimensional spaces. Expressions like these are especially useful when describing paths that are not easily defined by a single function, like spirals or curves with loops.

3D Graphing

3D graphing is crucial when visualizing vector-valued functions like our example. Unlike 2D graphs that depict only flat data, 3D graphs represent shapes and patterns in a fully three-dimensional space.
Tools such as Desmos or GeoGebra employ parametric equations to generate complex structures. When plotting \( \mathbf{r}(t) \), we use these systems to adjust parameter settings, such as the range of \( t \). This helps in capturing the dynamic nature of the curve efficiently.Graphing in 3D often involves:

• Inputting parametric equations
• Choosing a suitable range for \( t \)
• Manipulating the plot to alter viewing angles or scale

Effortlessly rotating and exploring the graph reveals insights into how surfaces shift across their respective axes. Understanding how components affect shape supports in-depth learning, allowing exploration of concepts like intersections and asymptotes.

Exponential Decay

Exponential decay describes a process where quantities decrease at a rate proportional to their current value. It’s represented using expressions like \( e^{-t} \), which appear in the parametric equations of our curve.
As \( t \) increases, \( e^{-t} \) diminishes, impacting the amplitude of \( x(t) \) and \( y(t) \). This effect produces a decaying spiral, tightening towards a focal point.In detail, exponential decay:

• Reduces the influence of the cosine and sine terms as \( t \) grows
• Causes components to shrink towards their limits
• In the \( z \)-component, \( 5 - 5e^{-t} \) starts from nearly zero and climbs towards a plateau at \( z=5 \)

This characteristic is seen in real-world phenomena like cooling temperatures or radioactive decay, making it important for students to grasp exponential behavior in various scientific and engineering disciplines.